Psychology Statistics 301 Exam 3 PDF

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Exam 3, notes, review...

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Psychology Statistics: Exam 3 Study Guide Topics: One- Sample T-test !Dependent sample T-test !Independent T- test !One-way ANOVA !Factorial ANOVA One Sample T- test

- One sample t-test asks the same question as the z test: Is the sample mean different from the population mean? Formula: S= estimated population standard deviation ! S = estimated standard deviation of sampling distribution, based on S ! M

Symbols: Greek letters: population parameters. Roman Letters: based on our sample Sample Population Sampling Distribution (from population)

Estimated population

Sampling distribution (from estimated population)

Standard Deviation

SD

σ

σM

S

SM

Variance

SD2

σ2

σM2

S2

SM2

Estimating Variability ! Sample statistic systematically differs from the population parameter because: - A bias

Psychology Statistics: Exam 3 Study Guide Topics: One- Sample T-test !Dependent sample T-test !Independent T- test !One-way ANOVA !Factorial ANOVA - One sample t-test asks the same question as the z test: Is the sample mean different from the population mean? Formula: S= estimated population standard deviation ! S = estimated standard deviation of sampling distribution, based on S ! M

Symbols: Greek letters: population parameters. Roman Letters: based on our sample Sample Population Sampling Distribution (from population)

Estimated population

Sampling distribution (from estimated population)

Standard Deviation

SD

σ

σM

S

SM

Variance

SD2

σ2

σM2

S2

SM2

Estimating Variability ! Sample statistic systematically differs from the population parameter because: - A bias

- Due to non-random sampling, poor design, etc. Sample variance (SD ) underestimates population variance σ - SD , calculated with n in the denominator, biased 1. Ok for our sample, not OK for the population - S , is calculated with n-1 in the denominator, is unbiased - OK for the population Formula: The new way: Denominator smaller, S bigger and better estimates σ 2

2

2

2

2

2

T- test Formula:

Test Statistic Distributions T statistic: compare to the t distribution - µ = 0, σ = 1 - extra parameter: df (n-1) the t distribution changes shape with df - smaller df= shorter distribution ‘heavier’ tails - larger df= taller distribution. ‘ skinnier; tails - infinite df= normal distribution T-table T table lists critical values - If t statistic > critical value, reject the null hypothesis - As df increases, critical value decreases Two measures taken from the same group of participants, or one

measure from each two paired participants Examples: “Before and after” experiments - How stressed are students before versus after taking a statistics exam? 2 conditions of 1 experiment - How far can people kick a football with their left versus their right foot? Comparison Distribution Dependent-samples t-test: means of difference scores - Scores are paired, so we build a distribution out of the means for (score 2- score 1) Null/ Research Hypotheses Very general null and research hypotheses (non-directional) H = the mean difference score is not equal to 0 H = the mean difference score is equal to 0 Very general null and research hypotheses (directional) H = the mean difference score is greater than (or less than). 0 H = the mean difference score is not greater than (or not less than) 0 Effect size Predicted effect size formula: A

0

A 0

µ = predicted mean of population of differences µ = 0 (H predicted mean of population of difference scores) σ = standard deviation of population of difference scores small d: 0.2 medium d: 0.5 large d: 0.8 Based on you sample formula: x# = sample mean difference score (we infer that x# = µ ) 0 = mean of comparison distribution (under H , µ = 0) 1 2

0

1

0

2

S = estimated standard deviation of population of difference scores (our estimate of σ based on our sample) One measure taken from each of two groups of participants - This experimental design is “between-subjects” Examples: 1. Do people with high stress or low stress get more sleep 2. Does involvement in extracellular activities have an effect on college GPA? 3. Is self-esteem higher for people with Facebook accounts or people without Facebook accounts? Comparison Distribution Independent-samples t-test: difference between means - Scores are not paired, so we build a distribution out of: (mean drawn from 1 sampling distribution) – (mean drawn from 2 sampling distribution) Null/ Research Hypotheses Very general null and research hypotheses (non-directional) H = the mean difference score is not equal to 0 H = the mean difference score is equal to 0 Very general null and research hypotheses (directional) H the mean for group 2 is greater than (less than) for group 1 H = the mean for group 2 is not greater than (not less than) the mean for group 1) Estimating Variability 1. Calculate S based on each sample This should look familiar: st

nd

A

0

A= 0

2

But now we use it twice, once for each sample Sample 1:

Sample 2:

2. Calculate the pooled estimate of population S How do we combine S and S We take a weighted average 3. Calculate the variance for each distribution of means We created a pooled estimate that encompasses both samples; now we divide it back up according to the size of each sample (so we know the variance of each sampling distribution) Before, we calculated S = S / n S = S /n , and S = S / n 4. Calculate the variance of the distribution of differences between means S =S +S 5. Calculate the standard deviation of distribution of differences between means: S (or S for short) S = √ S difference 2

pooled

2

2

1

2

2 M

2

2

M1

2

pooled

2

1

2

differnce

M2

2

2

2

pooled

2

M1

difference

M2

diff

2

difference

Formula: If H is true, difference in means will be small (low variance) If H is false, difference in means will be large (high variance) Numerator: variability between group means Denominator: variability within group means If numerator > denominator, likely to be a real difference between groups Analysis of Variance (ANOVA) The F test compares the ration of variability between 3+ groups to the variability within the groups Formula: 0

0

H = All group means are equal - µ1 = µ2 = ... = µ H = Not all groups means are equal - But, doesn’t specify which means are different from which S - Variability due to chance factors (beyond experimental control) - Also called “error variance” (S S S ) - Pooled estimate of variability across different groups - This is the same whether H is true or false One way to calculate S for each group 1. Calculate S for each group 2. Take the average F Distribution Z distribution: it was like having infinite df - Normal distribution T distribution: one measure of df - Approached normal distribution when df= infinity F distribution: two measures of df - Nowhere near normally distributed - F greater than or equal to (not symmetric) - Changes shape with both df F table F tables are organized by df F critical values in a table work just like t critical values - If your F score > F critical value, reject H - If your F score < F critical value, do not reject H NHST using ANOVA Step 1: Restate the question as a research hypothesis and a null hypothesis about the populations - Define groups; make non-directional research hypothesis Step 2: Determine the characteristics of the comparison 0

last

A

2

within

2

2

w’

2

win’

error

0

2

2

0

0

distribution - F distribution with (df , df ) df Step 3: Determine the cutoff sample score on the comparison distribution at which the null hypothesis should be rejected Look this up in the F table; based on α, df , df Step 4: Determine your sample’s score on the comparison distribution - Conduct ANOVA, is F statistic > F critical value? Step 5: Decide whether to reject the null hypothesis Step 6: Interpret your results - Remember, your hypothesis is non-directional ANOVA Results ANOVAs tell us if at least one group is different from the others If we reject H , what do we do next? - We need some way of comparing pairs of groups to find out which one is different There are two way to achieve this: - Planned contrasts (difference expected beforehand) - Post-hoc tests (difference not necessarily expected at all) Post HOC Tests Only rule: you must reject H in order to do a post hoc test - One option is to reduce α using Bonferroni’s correction New α = α/ #tests - Use a new, smaller alpha for each test to make sure that there is still only a 5% total chance of Type 1 error. - e.g. For an ANOVA with 5 groups, there are 10 pairwise contrasts. We can test each. Using t-tests at α = (0.0.5/10) = 0.05 - “Bonferroni-corrected. T-tests” ANOVAS: Two Designs - Between-subjects ANOVAs: each person in only one group, one measurement per person - Within-subjects ANOVAs: each person in each group, multiple measurements per person between

within

between

0

0

within

Investigate the effects of more than IV - Example: does improvement in a math test depend on the drug dose and/or the time of day that it is taken Factorial ANOVA (> 1 independent variable tested) Two-Way ANOVA Two-way ANOVAs have 3 hypothesis and 3 F-tests - Main effects: whether there are differences between levels of a single variable (same as before) - variable A: Dose affects improvement - Variable B: time of day affects improvement - Interaction effects: whether the variables together have different effects from the variables individually Written as A x B, or A-by-B interaction SS Two-way ANOVA - Calculating SS for 2 IVs (A,B) and interaction (AB) Formula: - A, B, and AB are all between-subject factors - Error is within-subject (what we used to call SS ) DF Two-way ANOVA: - Calculating df for 2 IVs (A, B) and interaction (AB) Formula: w

- A, B, and AB are all between-subject factors - Error is within-subjects (what we used to call df ) For each variable: 2

For the interaction:

For the error term:

For the total df: Look At Exam 3 Page on Factorial ANOVA Power point.

Exam 3 Practice Set 1. What statistical test would we use if we measured sleepiness of each student in this class before this review session began and again after this review session ended? A. z-test B. One-sample t-test D. Independent-samples t-test 2. The F-statistic is a ratio between ____ and ____? F = A. Between group variance (MS ) / Within group variance (Ms ) B. Within group variance (Ms ) / Between group variance (MS ) C. Sum of squares total (SS ) / degrees of freedom total (df ) D. None of the above between

within

between

total

total

3. If we are trying to get the largest F-statistic (and increase our chances of rejecting the null hypothesis), which variance do we want to maximize? In other words, which variance is good and which is bad? 4. Which of the following statements about a one-way ANOVA is FALSE? A. Hypotheses must be non-directional B. F-statistics can only be positive C. A significant result only tells you that at least one group is different from the rest D. None of the above; all are true

5. Bob conducts an ANOVA and gets a significant result. He wants to conduct 6 follow-up t-tests but knows that he should use a correction so that he doesn’t inflate his alpha. If Bob’s original α=.05 and he uses the Bonferroni correction, what would the new alpha level be for his t-tests? A. .05 B. .30 C. .01 D. .008 6. Jason thinks that people post on Facebook on Saturdays more than any other day of the week. He samples 2 and found that on average they post on Saturdays (SS = compared to a population average of 3 times the other days of the week. Calculate the tstatistic for this one-sample t-test. A. 3.45 B. 4.68 D. 6.82 7. Calculate the t-statistic for a one-sample t-test using the following data: [15 12 16 9] against a population mean of µ = 18. B. 3.16 C. -0.5 D. 0.5 8. Calculate the t-statistic in a paired-sample t-test for happiness levels before and after an elementary school field trip to the zoo:

Before = [ 7 4 6 ] After = [ 9 7 10] A. 2.11 B. 3.92 C. 4.57 D. 5.19 !

9. Calculate S given the following information: S = 5.5, n = 5 S = 4.6, n = 7 A. 1.30 B. 2.32 C. 3.45 D. 4.88 diff

2

1

1

2

2

2

10. Answer the following:

a. How many groups are in this experiment? b. How many total subjects participated in this experiment? c. Calculate MS . d. Calculate MS . e. Calculate the F-statistic. f. What is the effect size for this experiment? between within

11. Tasha wants to know whether amount of sleep varies by year in college (freshman, sophomore, junior, senior). She collects average sleep from 100 students (25 in each

year). SS = 36. SS = 320. a. What is the df ? b. What is the df ? c. Calculate MS . d. Calculate MS . e. Calculate the F-statistic. between

within

between within

between within

For 12 – 15, use the question from 12: 12. Laura wants to test several factors that might influence hunger levels, including: gender (male, female), current stress level (low, medium, high), and amount of sleep the previous night (6 hours). What statistical test should Laura use? A. One-way ANOVA B. Factorial ANOVA C. Dependent t-test D. Independent t-test 13. How many independent variables does Laura have in her experiment? A. 2 C. 8 D. 11 14. How would you describe this ANOVA? B. 3-by-8 ANOVA C. 2-by-3-by-4 ANOVA D. 3-by-2-by-3-by-3 ANOVA 15. How else could you describe this ANOVA?

A. Between subjects design B. Within subjects design C. Mixed design D. Repeated measures design 16. What effect does the figure below show:

A. Main effect of dose B. Main effect of time D. Both A and B 17. What effect does the figure below show:

A. Main effect of dose B. Main effect of time C. Interaction effect D. Both A and B

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